Solution
= Solution
The <free particle> has <Hamiltonian operator> $H=P^2/(2m)$, with <momentum operator> $P=-i\hbar\partial_x$. Its time-dependent <Schrödinger equation> is
$$
\boxed{i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}},\qquad
\Psi(t,x+2\pi R)=\Psi(t,x).
$$
Here $x$ measures arc length for the <particle on a circle>, so the <wavefunction normalization> uses $dx$, rather than $d(x/R)$.